Cartesian differential categories

R.F. Blute, J.R.B. Cockett and R.A.G. Seely

This paper revisits the authors' notion of a differential
category from a different perspective. A differential category is an
additive symmetric monoidal category with a comonad (a "coalgebra
modality") and a differential combinator. The morphisms of a
differential category should be thought of as the linear maps; the
differentiable or smooth maps would then be morphisms of the coKleisli
category. The purpose of the present paper is to directly axiomatize
differentiable maps and thus to move the emphasis from the linear notion
to structures resembling the coKleisli category. The result is a setting
with a more evident and intuitive relationship to the familiar notion of
calculus on smooth maps. Indeed a primary example is the category
whose objects are Euclidean spaces and whose morphisms are smooth maps.

A Cartesian differential category is a Cartesian left additive
category which possesses a Cartesian differential operator. The
differential operator itself must satisfy a number of equations, which
guarantee, in particular, that the differential of any map is `"linear"
in a suitable sense.

We present an analysis of the basic properties of Cartesian
differential categories. We show that under modest and natural
assumptions, the coKleisli category of a differential category is
Cartesian differential. Finally we present a (sound and complete) term
calculus for these categories which allows their structure to be
analysed using essentially the same language one might use for
traditional multi-variable calculus.